Question

give the center and radius of the circle described by the equation and graph the equation. use the graph to identify the domain and range.

$(x + 4)^{2}+(y + 3)^{2}=4$
the center is $(-4,-3)$
(type an ordered pair. simplify your answer.)
the radius is $2$
(type an integer or a simplified fraction.)
graph the circle.

Explanation:

Step1: Recall the standard form of a circle equation

The standard form of a circle equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius.

Step2: Identify the center

For the equation \((x + 4)^2+(y + 3)^2=4\), we have \(h=-4\) and \(k = - 3\). So the center \((h,k)=(-4,-3)\).

Step3: Identify the radius

Since \(r^2 = 4\), then \(r=\sqrt{4}=2\).

Step4: Find the domain

The left - most \(x\) value is \(h - r=-4-2=-6\) and the right - most \(x\) value is \(h + r=-4 + 2=-2\). So the domain is \([-6,-2]\).

Step5: Find the range

The bottom - most \(y\) value is \(k - r=-3-2=-5\) and the top - most \(y\) value is \(k + r=-3 + 2=-1\). So the range is \([-5,-1]\).

Answer:

The center is \((-4,-3)\), the radius is \(2\), the domain is \([-6,-2]\) and the range is \([-5,-1]\).